2 Answers
2

I would think $\gneq$ means exactly the same as $>$, i.e. it would mean greater than and not equal to (while the symbol $\geq$ means greater than or equal to). But of course there may be some specialized use where it doesn't mean this though; everything depends on context.

$$n\gneq 3 \iff n>3 \iff n\text{ is greater than }3$$
and, because $n$ is an integer in this context, we can also say that
$$n\gneq 3\iff n\geq 4.$$

As Rasmus points out below, the analogous notations with set inclusion, $\subset$ vs. $\subsetneq$, unfortunately do not mean the same in general; many authors use $A \subset B$ to mean "$A$ is a subset of $B$, and could be equal to $B$". An unambiguous alternative to express that would be to write $\subseteq$.

This is clearly the general usage. It might be worthwile to point out that the set-theoretic notation $\subset$ is not universally used for proper inclusion. To be on the safe side, most people only use $\subseteq$ and $\subsetneq$.
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RasmusDec 8 '11 at 15:30

yes we can write $x^2+1 \gt 0$ but when we write $x^2+1 \gneq 0$ we emphasis that it can't be zero. I mean the point is emphasizing because I've seen this sign when it was necessary to not be equal. For example when the statement is in the denominator of a fraction and we want to emphasis that it can't be zero. And Thanks for the +1!
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BardiaDec 8 '11 at 17:23

@Bardia : Very nice answer !! , convinced with it, why can't you include the same application of fractions in your answer
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IyengarDec 8 '11 at 17:30

@TheChaz : dimunitive ? , I never know the person before, but just wanted to tell so, but if you mind keeping that I better delete it
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IyengarDec 8 '11 at 17:33

@iyengar: fair enough! I should have included that example too.
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BardiaDec 8 '11 at 17:42